The Quantization of Musical Time: A Connectionist Approach

Abstract

Musical time can be considered to be the product of two time scales: the discrete time intervals of a metrical structure and the continuous time scales of tempo changes and expressive timing (Clarke 1987a). In musical notation both kinds are present, although the notation of continuous time is less developed than that of metric time (often just a word like "rubato" or "accelerando" is notated in the score). In the experimental literature, different ways in which a musician can add continuous timing changes to the metrical score have been identified. There are systematic changes in certain rhythmic forms: for example, shortening triplets (Vos and Handel 1987) and timing differences occurring in voice leading with ensemble playing (Rasch 1979). Deliberate departures from metricality, such as rubato, seem to be used to emphasize musical struc- ture, as exemplified in the phrase-final lengthening principle formalized by Todd (1985). In addition to these effects, which are collectively called expressive timing, there are nonvoluntary effects, such as random timing errors caused by the limits in the accuracy of the motor system (Shaffer 1981) and errors in mental time-keeping processes (Vorberg and Hambuch 1978). These effects are generally rather small - in the order of 10-100 msec. To make sense of most musical styles, it is necessary to separate the discrete and continuous components of musical time. We will call this process of separation quantization, although the term is generally used to reflect only the extraction of a metrical score from a musical performance.